Regularity in Linear Degenerations of Flag Variety

TL;DR

This paper studies the smoothness, singularities, and stratification of linear degenerations of flag varieties, establishing normality of irreducible degenerations and classifying smooth cases.

Contribution

It classifies smooth linear degenerations of flag varieties, analyzes their singular loci, and introduces a new stratification of the representation space.

Findings

Smooth linear degenerations are classified.

Irreducible degenerations are shown to be normal.

Estimates for the dimension of the singular locus are provided.

Abstract

In this article we investigate the regularity properties of linear degenerations of flag varieties. We classify the linear degenerations of (partial) flag varieties that are smooth. Furthermore, we study the singular locus of irreducible degenerations and provide estimates for its dimension. We also introduce a new stratification of the total space of representations. Within each stratum, we identify the loci corresponding to flat and flat irreducible degenerations. As a consequence of our results, we show that irreducible linear degenerations are normal varieties.